Travel-Time Tomography: From Rays to Inversion

Contents

Travel-Time Tomography: From Rays to Inversion#

Learning Objectives#

After this lecture, students should be able to:

  • Explain how travel-time residuals encode Earth structure

  • Write travel-time tomography as a linear inverse problem

  • Construct the tomography matrix (G) from ray geometry

  • Explain why the problem is ill-posed

  • Describe the role of regularization

  • Understand why ray bending makes tomography nonlinear

This lecture builds directly on:

  • ray tracing and ray bending

  • Snell’s law and Fermat’s principle

  • linear inverse problems

1. What Are We Measuring? (5 min)#

On the board#

Draw:

  • a source

  • a receiver

  • a ray path

  • a layered or heterogeneous medium

Write:

\[ t_i^{\text{obs}} \quad\text{and}\quad t_i^0 \]

Define the data:

\[ d_i \equiv \delta t_i = t_i^{\text{obs}} - t_i^0 \]

Say explicitly#

  • We do not invert absolute travel times

  • We invert residuals relative to a reference model

Physical meaning#

  • (\(\delta t > 0\)): slower than reference

  • (\(\delta t < 0\)): faster than reference

Concept check (ask students)#

If all arrivals at one station are late, what could that mean?

(velocity anomaly? station term? origin time?)

2. Forward Problem: Travel Time as an Integral (5–7 min)#

On the board#

Start from ray theory:

\[ t = \int_{\Gamma} u(\mathbf{x})\, ds \quad\text{with}\quad u(\mathbf{x}) = \frac{1}{v(\mathbf{x})} \]

Split the model:

\[ u(\mathbf{x}) = u_0(\mathbf{x}) + \delta u(\mathbf{x}) \]

Linearize:

\[ \delta t = \int_{\Gamma} \delta u(\mathbf{x})\, ds \]

Critical assumption (circle this)#

Rays are frozen in the reference model

\[ \Gamma \approx \Gamma_0 \]

Say out loud#

  • This is a first-order (Born-like) approximation

  • Valid for small perturbations

  • Ray bending enters later

Concept check#

Why does Fermat’s principle justify freezing the ray path at first order?

3. Discretization: From Integrals to Sums (8 min)#

On the board#

Draw:

  • the domain divided into rectangular cells

  • one ray crossing several cells

Write:

\[ \delta t_i \approx \sum_{j=1}^{M} L_{ij}\,\delta u_j \]

Where:

  • (\(\delta u_j\)): slowness perturbation in cell \(j\)

  • (\(L_{ij}\)): path length of ray \(i\) in cell \(j\)

Matrix form (box this)#

\[ \mathbf{d} = \mathbf{G}\mathbf{m} \]

with:

  • (\(d_i = \delta t_i\))

  • (\(m_j = \delta u_j\))

  • (\(G_{ij} = L_{ij}\))

Emphasize#

  • All the physics is in \(G\)

  • \(G\) is geometry + ray paths

  • The inverse problem is bookkeeping, not magic

Concept check#

What happens to \(G\) in a cell that no ray crosses?

4. Why Tomography Is Hard (Ill-posedness) (6–8 min)#

On the board#

Write the least-squares problem:

\[ \min_{\mathbf{m}} \|\mathbf{Gm} - \mathbf{d}\|_2^2 \]

Then the normal equations:

\[ (\mathbf{G}^T\mathbf{G})\mathbf{m} = \mathbf{G}^T\mathbf{d} \]

Discuss three failure modes#

  1. Poor coverage

    • columns of \(G\) nearly zero

  2. Similar ray paths

    • rows nearly linearly dependent

  3. Limited ray angles

    • directional smearing

Draw elongated resolution kernels aligned with ray directions.

Key message (underline)#

Tomography is limited by ray geometry, not by inversion technique.

5. Regularization: Making the Problem Solvable (8 min)#

Damping (minimum-norm)#

Write:

\[ \min_{\mathbf{m}} \|\mathbf{Gm}-\mathbf{d}\|^2 + \lambda^2 \|\mathbf{m}\|^2 \]

Say:

  • favors small-amplitude anomalies

  • fills gaps with zeros

Smoothness (minimum-roughness)#

Write:

\[ \min_{\mathbf{m}} \|\mathbf{Gm}-\mathbf{d}\|^2 + \lambda^2 \|\mathbf{L}\mathbf{m}\|^2 \]

where \(\mathbf{L}\) is a discrete Laplacian.

Say:

  • favors smooth Earth

  • fills gaps with interpolation

Ask students#

Which regularization would you trust more in the mantle? In sedimentary basins?

6. P vs S Tomography (2–3 min)#

On the board#

Write:

\[ \delta t_P = \int \delta u_P\, ds \quad\text{and}\quad \delta t_S = \int \delta u_S\, ds \]

Emphasize#

  • Mathematics is identical

  • Differences are:

    • ray coverage

    • picking quality

    • sensitivity to fluids

Key point#

P and S tomography differ in data, not in formulation.

7. Why Ray Bending Makes Tomography Nonlinear (5–7 min)#

On the board#

Draw:

  • straight ray through a low-velocity anomaly

  • bent ray avoiding slow region

Write:

\[ t(u) = \int_{\Gamma(u)} u(\mathbf{x})\, ds \]

Say clearly:

  • \(t\) depends on \(u\)

  • and \(\Gamma\) depends on \(u\)

Linearization with iteration#

At iteration \(k\):

\[ \delta t^{(k)} \approx \int_{\Gamma^{(k)}} \delta u^{(k)}\, ds \]

Algorithm sketch (numbered)#

  1. Start with \(u^{(0)}\)

  2. Trace rays \(\Gamma^{(0)}\)

  3. Build \(G^{(0)}\)

  4. Solve for \(\delta u^{(0)}\)

  5. Update \(u^{(1)} = u^{(0)} + \alpha \delta u^{(0)}\)

  6. Repeat

Circle:

\[ \text{model} \rightarrow \text{rays} \rightarrow G \rightarrow \text{update} \]

8. Connection to the Notebook#

Tell students#

  • Part A notebook = straight-ray tomography

  • Part B notebook = iterative bent-ray tomography

  • PyKonal solves:

    • eikonal equation

    • ray tracing via \(\nabla T\)

Prediction question#

What failure mode of straight-ray tomography does ray bending fix best?

9. Take-Home Messages (Final 2 min)#

  • Travel-time tomography is a geometric inverse problem

  • \(G\) is built from ray paths

  • Ill-posedness is unavoidable → regularization is physics

  • Straight-ray tomography is linear but approximate

  • Bent-ray tomography is more accurate but nonlinear

  • Resolution is controlled by coverage, not clever math

What Comes Next#

  • Joint inversion with:

    • event locations

    • origin times

    • station terms

  • Finite-frequency sensitivity kernels

  • Comparison with surface-wave tomography

If you want next, I can:

  • convert this into ADA-compliant slides with figures + alt-text

  • add marginal instructor notes (what to emphasize live vs skip)

  • write a one-page student summary that complements the notebook without duplicating it

Travel-Time Tomography: From Rays to Inversion#

Learning Objectives#

After this lecture, students should be able to:

  • Explain how travel-time residuals encode Earth structure

  • Write travel-time tomography as a linear inverse problem

  • Construct the tomography matrix (G) from ray geometry

  • Explain why the problem is ill-posed

  • Describe the role of regularization

  • Understand why ray bending makes tomography nonlinear

This lecture builds directly on:

  • ray tracing and ray bending

  • Snell’s law and Fermat’s principle

  • linear inverse problems

1. What Are We Measuring? (5 min)#

On the board#

  • a source

  • a receiver $\( t^{\text{obs}}_i \quad\text{and}\quad t^0_i \)$

  • a layered or heterogeneous medium

\[ d_i \equiv \delta t_i = t^{\text{obs}}_i - t^0_i \]

\quad\text{and}\quad t^0_i

Define the data: $\( d_i \equiv \delta t_i = t^{\text{obs}}_i - t^0_i \)$

Say explicitly#

  • We do not invert absolute travel times

  • We invert residuals relative to a reference model

Physical meaning#

  • ( \(\delta t > 0\) ): slower than reference

Concept check (ask students)#

If all arrivals at one station are late, what could that mean?

(velocity anomaly? station term? origin time?)#

2. Forward Problem: Travel Time as an Integral (5–7 min)#

\[ t = \int_{\Gamma} u(\mathbf{x})\, ds \quad\text{with}\quad u(\mathbf{x}) = \frac{1}{v(\mathbf{x})} \]

On the board#

\[ u(\mathbf{x}) = u_0(\mathbf{x}) + \delta u(\mathbf{x}) \quad\text{with}\quad u(\mathbf{x}) = \frac{1}{v(\mathbf{x})} \]

Split the model: $\( \delta t = \int_{\Gamma} \delta u(\mathbf{x})\, ds u(\mathbf{x}) = u_0(\mathbf{x}) + \delta u(\mathbf{x}) \)$

Linearize \(\delta t\): $\( \Gamma \approx \Gamma_0 \)$

Critical assumption (circle this)#

Rays are frozen in the reference model $\( \Gamma \approx \Gamma_0 \)$

Say out loud#

  • Valid for small perturbations

  • Ray bending enters later

Concept check#

Why does Fermat’s principle justify freezing the ray path at first order?

3. Discretization: From Integrals to Sums (8 min)#

Draw:

\[ \delta t_i \approx \sum_{j=1}^{M} L_{ij}\,\delta u_j \]

  • one ray crossing several cells

Write: $\( \delta t_i \approx \sum_{j=1}^{M} L_{ij},\delta u_j \)\( Where: \)\( \mathbf{d} = \mathbf{G}\mathbf{m} \)$

  • \( \delta u_j \): slowness perturbation in cell (j)

  • \( L_{ij} \): path length of ray (i) in cell (j)

Matrix form (box this)#

\[ \mathbf{d} = \mathbf{G}\mathbf{m} \]

with:

  • \( d_i = \delta t_i \)

  • \( m_j = \delta u_j \)

  • \( G_{ij} = L_{ij} \)

Emphasize#

  • (G) is geometry + ray paths

  • The inverse problem is bookkeeping, not magic

Concept check#

What happens to (G) in a cell that no ray crosses?

4. Why Tomography Is Hard (Ill-posedness) (6–8 min)#

Goal is to minimize the difference between observed and predicted data: $\( \min_{\mathbf{m}} \|\mathbf{Gm} - \mathbf{d}\|_2^2 \)\( \)G\( is not square and full rank, so we calculate the *normal equation*: \)\( \mathbf{G}\mathbf{m} = \mathbf{d} \)$

\[ (\mathbf{G}^T\mathbf{G})\mathbf{m} = \mathbf{G}^T\mathbf{d} \]

Then the normal equations: $\( \mathbf{m} = (\mathbf{G}^T\mathbf{G})^{-1}\mathbf{G}^T\mathbf{d} \)$

Discuss three failure modes#

  1. Poor coverage

    • columns of (G) nearly zero

  2. Similar ray paths

    • rows nearly linearly dependent

  3. Limited ray angles

Draw elongated resolution kernels aligned with ray directions.

Key message (underline)#

Tomography is limited by ray geometry, not by inversion technique.

5. Regularization: Making the Problem Solvable (8 min)#

\[ \min_{\mathbf{m}} \|\mathbf{Gm}-\mathbf{d}\|^2 + \lambda^2 \|\mathbf{m}\|^2 \]

Damping (minimum-norm)#

Write: $\( \min_{\mathbf{m}} |\mathbf{Gm}-\mathbf{d}|^2 + \lambda^2 |\mathbf{m}|^2 \)$ Say: Damping favors small-amplitude anomalies

Smoothness (minimum-roughness)#

Write: $\( \min_{\mathbf{m}} |\mathbf{Gm}-\mathbf{d}|^2 + \lambda^2 |\mathbf{L}\mathbf{m}|^2 \)$

where \( \mathbf{L} \) is a discrete Laplacian.

  • favors smooth Earth

  • fills gaps with interpolation

6. P vs S Tomography (2–3 min)#

On the board#

Write: $\( \delta t_P = \int \delta u_P, ds \quad\text{and}\quad \delta t_S = \int \delta u_S, ds \)$

Emphasize#

  • Mathematics is identical

    • ray coverage

    • picking quality

    • sensitivity to fluids

Key point#

P and S tomography differ in data, not in formulation.

\[ t(u) = \int_{\Gamma(u)} u(\mathbf{x})\, ds \]

Draw:

  • straight ray through a low-velocity anomaly

  • bent ray avoiding slow region

Write: $\( t(u) = \int_{\Gamma(u)} u(\mathbf{x}), ds \)$ Say clearly:

\[ \delta t^{(k)} \approx \int_{\Gamma^{(k)}} \delta u^{(k)}\, ds \]

  • and \(\Gamma\) depends on \(u\)

Linearization with iteration#

At iteration (k): $\( \delta t^{(k)} \approx \int_{\Gamma^{(k)}} \delta u^{(k)}, ds \)$

\[ ext{model} \rightarrow \text{rays} \rightarrow G \rightarrow \text{update} \]

  1. Trace rays \(\Gamma^{(0)}\)

  2. Solve for \(\delta u^{(0)}\)

  3. Update \(u^{(1)} = u^{(0)} + \alpha \delta u^{(0)}\)

  4. Repeat

Circle: $\( \text{model} \rightarrow \text{rays} \rightarrow G \rightarrow \text{update} \)$

8. Connection to the Notebook#

Tell students#

  • Part A notebook = straight-ray tomography

  • PyKonal solves:

    • eikonal equation

    • ray tracing via \(\nabla T\)

Prediction question#

What failure mode of straight-ray tomography does ray bending fix best?

9. Take-Home Messages (Final 2 min)#

  • Travel-time tomography is a geometric inverse problem

  • (G) is built from ray paths

  • Ill-posedness is unavoidable → regularization is physics

  • Straight-ray tomography is linear but approximate

  • Bent-ray tomography is more accurate but nonlinear

  • Resolution is controlled by coverage, not clever math

  • Joint inversion with:

    • event locations

    • origin times

    • station terms

  • Finite-frequency sensitivity kernels

  • Comparison with surface-wave tomography

Contents